Question

If the sound level is $60dB$ close to a speaker that has an area of $120c{m^2}$. The acoustic power output of the speaker is $12 \times {10^{ - x}}$ watt. Find the value of $x$.

Answer 1

Hint:Acoustic power is the rate at which the sound wave is reflected, transmitted, emitted or received per time. Intensity is defined to be the power per unit area carried by a wave so, the acoustic power output of the speaker depends on the area of the speaker and the sound intensity. Find the value of intensity of the sound wave and covert all the terms to the required dimension for the further steps.

Complete Step by step solution: We know the loudness of the sound is given by the equation,
\[L = \log \left( {\dfrac{I}{{{I_0}}}} \right)\]
Where, $I$ is the sound intensity in watt per square meter
The value of ${I_0}$ is given by, ${I_0} = {10^{ - 12}}W{m^{ - 2}}$
From the equation, we get,
$\dfrac{I}{{{I_0}}} = {10^L}$…………………………………….. (1)
In the question, it is given that the value of loudness of the speaker is $60dB$
That is, $L = 60dB$
By using mathematical conversions, we get,
\[L = 6bel\]
And we know ${I_0} = {10^{ - 12}}W{m^{ - 2}}$
Applying these values to the equation (1)
We get, $\dfrac{I}{{{I_0}}} = {10^L}$
That is, \[\dfrac{I}{{{I_0}}} = {10^6}\]
Applying the value of ${I_0}$,
We, get, \[\dfrac{I}{{{{10}^{ - 12}}}} = {10^6}\]
Shifting the value of ${I_0}$ to the right hand side of the equation we get,
\[ \Rightarrow I = {10^6} \times {10^{ - 12}}\]
\[ \Rightarrow I = {10^{ - 6}}W{m^{ - 2}}\]

From the question itself, we know the area of the speaker, $A = 120c{m^2}$
By using simple mathematical conversions, we get are of the speaker,
$A = 120 \times {10^{ - 4}}{m^2}$
It is known that the acoustic power output of the speaker is equal to the product of area of the speaker and the sound intensity.
That is, $P = A \times I$
By applying the known values of the terms in the above equation we get,
$P = {10^{ - 6}} \times 120 \times {10^{ - 4}}$
$ \Rightarrow 120 \times {10^{ - 10}}$
$ \Rightarrow 12 \times {10^{ - 9}}W$
Now we got the value of the acoustic power output of the speaker,
$P = 12 \times {10^{ - 9}}W$
Now we take a look into the question in which we are asked to find the value of $x$ which is after the negative sign.

Now we get $x = 9$ is the final answer.

Note:it is important to note the units given in the question, we need to convert the units to the desired dimension to calculate these types of questions. Otherwise the chance making mistakes are higher.